I'm looking for properties (P) such that you would assume that there are infinitely many natural numbers with property (P) (for example because there are very large numbers with that property) but where it turns out that there are only finitely many.

EDIT: Note that the eventual counterexamples question asks for (P) such that the smallest $n$ with property (P) is large; the current question asks for (P) such that the largest $n$ with property (P) is large.

Not sure I agree. 115132219018763992565095597973971522401 is the last $n$-digit number equal to the sum of the $n$th powers of its digits; I don't see it as an eventual counterexample to anything.
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Gerry MyersonJul 6 '12 at 12:44

2

I am going to start a meta thread to ask for reopening.
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Gerry MyersonJul 8 '12 at 9:09

I don't think this one should really count; sporadic groups are by definition those that don't fall into an infinite family. If there were infinitely many, they wouldn't be sporadic. There's a largest sporadic group because that's what "sporadic group" means.
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Harry AltmanAug 3 '12 at 22:20

@Harry, I don't see any reason why there couldn't have been infinitely many sporadic (finite simple) groups. Why can't you have infinitely many things that don't fall into a family?
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Gerry MyersonAug 4 '12 at 12:21

115132219018763992565095597973971522401 is the 88th and last $n$-digit number equal to the sum of the $n$-th powers of its digits. Such numbers are sometimes called "narcissistic numbers", sometimes called "Armstrong numbers".

A good start might be finite sequences in OEIS. The search keyword: fini returns
4660 results including Gerry's narcissistic number and left-truncatable prime.
Another example is A080601 Number of positions that the 3 X 3 X 3 Rubik cube puzzle can be in after exactly n moves with largest term $91365146187124313$.

The search can be automated filtering only large numbers.

Probably you are not interested in this, but artificial solutions can be constructed easily:

I don't believe my answer misses the point; it simply addresses the question some had expressed about a connection between the questions. But now I shall say more: in my answer to the other question (mathoverflow.net/questions/15444/…), I explain my perspective that the essence of these questions is about giving very short descriptions of very large numbers. And those remarks seem to apply equally here...
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Joel David HamkinsJul 9 '12 at 14:25

145068705885714027751024986638948255675293681033906152595410365894525140785818
941353479613216850845570730091684198720104504503331710760582412207588159919853
0284489710255042635927388160000000000
is the 245th and last 6-perfect number (that is, the sum of its divisors is 6 times the number).

3608528850368400786036725 is the 20457th and last number every prefix of which is divisible by the number of digits of the prefix (that is, 3 is divisible by 1, 36 is divisible by 2, 360 is divisible by 3, etc.).